YES
We show the termination of the TRS R:

  half(|0|()) -> |0|()
  half(s(|0|())) -> |0|()
  half(s(s(x))) -> s(half(x))
  lastbit(|0|()) -> |0|()
  lastbit(s(|0|())) -> s(|0|())
  lastbit(s(s(x))) -> lastbit(x)
  conv(|0|()) -> cons(nil(),|0|())
  conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))


-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: half#(s(s(x))) -> half#(x)
p2: lastbit#(s(s(x))) -> lastbit#(x)
p3: conv#(s(x)) -> conv#(half(s(x)))
p4: conv#(s(x)) -> half#(s(x))
p5: conv#(s(x)) -> lastbit#(s(x))

and R consists of:

r1: half(|0|()) -> |0|()
r2: half(s(|0|())) -> |0|()
r3: half(s(s(x))) -> s(half(x))
r4: lastbit(|0|()) -> |0|()
r5: lastbit(s(|0|())) -> s(|0|())
r6: lastbit(s(s(x))) -> lastbit(x)
r7: conv(|0|()) -> cons(nil(),|0|())
r8: conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))

The estimated dependency graph contains the following SCCs:

  {p3}
  {p1}
  {p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: conv#(s(x)) -> conv#(half(s(x)))

and R consists of:

r1: half(|0|()) -> |0|()
r2: half(s(|0|())) -> |0|()
r3: half(s(s(x))) -> s(half(x))
r4: lastbit(|0|()) -> |0|()
r5: lastbit(s(|0|())) -> s(|0|())
r6: lastbit(s(s(x))) -> lastbit(x)
r7: conv(|0|()) -> cons(nil(),|0|())
r8: conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        conv#_A(x1) = max{1, x1 - 13}
        s_A(x1) = x1 + 27
        half_A(x1) = max{10, x1 - 16}
        |0|_A = 8
        lastbit_A(x1) = max{36, x1 - 55}
        conv_A(x1) = max{9, x1 - 91}
        cons_A(x1,x2) = max{7, x1 - 84, x2 - 37}
        nil_A = 0
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          conv#_A(x1) = max{0, x1 - 5}
          s_A(x1) = max{2, x1}
          half_A(x1) = max{4, x1 - 3}
          |0|_A = 1
          lastbit_A(x1) = max{2, x1 - 3}
          conv_A(x1) = max{0, x1 - 5}
          cons_A(x1,x2) = max{0, x1 - 1, x2 - 3}
          nil_A = 0
          conv#!_A(x1) = max{5, x1 - 5}
          s!_A(x1) = max{0, x1 - 8}
          half!_A(x1) = 4
          |0|!_A = 1
          lastbit!_A(x1) = max{2, x1 - 19}
          conv!_A(x1) = max{3, x1 - 16}
          cons!_A(x1,x2) = max{0, x1 - 9, x2 - 14}
          nil!_A = 0
  
    partial status:
  
      pi(conv#) = []
      pi(s) = []
      pi(half) = []
      pi(|0|) = []
      pi(lastbit) = []
      pi(conv) = []
      pi(cons) = []
      pi(nil) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: half#(s(s(x))) -> half#(x)

and R consists of:

r1: half(|0|()) -> |0|()
r2: half(s(|0|())) -> |0|()
r3: half(s(s(x))) -> s(half(x))
r4: lastbit(|0|()) -> |0|()
r5: lastbit(s(|0|())) -> s(|0|())
r6: lastbit(s(s(x))) -> lastbit(x)
r7: conv(|0|()) -> cons(nil(),|0|())
r8: conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        half#_A(x1) = max{4, x1 + 2}
        s_A(x1) = max{23, x1 + 3}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          half#_A(x1) = 0
          s_A(x1) = max{14, x1 - 16}
          half#!_A(x1) = 0
          s!_A(x1) = max{3, x1 - 49}
  
    partial status:
  
      pi(half#) = [1]
      pi(s) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: lastbit#(s(s(x))) -> lastbit#(x)

and R consists of:

r1: half(|0|()) -> |0|()
r2: half(s(|0|())) -> |0|()
r3: half(s(s(x))) -> s(half(x))
r4: lastbit(|0|()) -> |0|()
r5: lastbit(s(|0|())) -> s(|0|())
r6: lastbit(s(s(x))) -> lastbit(x)
r7: conv(|0|()) -> cons(nil(),|0|())
r8: conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))

The set of usable rules consists of

  r1, r2, r4

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        lastbit#_A(x1) = x1 + 3
        s_A(x1) = x1 + 4
        half_A(x1) = x1 + 7
        |0|_A = 0
        lastbit_A(x1) = max{5, x1}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          lastbit#_A(x1) = 0
          s_A(x1) = max{9, x1 - 14}
          half_A(x1) = max{13, x1 - 31}
          |0|_A = 0
          lastbit_A(x1) = max{0, x1 - 24}
          lastbit#!_A(x1) = 0
          s!_A(x1) = max{8, x1 + 7}
          half!_A(x1) = max{15, x1 - 8}
          |0|!_A = 32
          lastbit!_A(x1) = 8
  
    partial status:
  
      pi(lastbit#) = [1]
      pi(s) = [1]
      pi(half) = [1]
      pi(|0|) = []
      pi(lastbit) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.